Method For Recognition Of Acyclic Instruction Patterns

ABSTRACT

A method which determines by an optimizing compiler whether any variable in the given program equals to the given acyclic mathematical function f(x,y, . . . ) applied to the given variables x, y, . . . in the program. In one embodiment, the method includes expressing the bits of the value of the function f(x,y, . . . ) as a Boolean function of the bits of the inputs x, y, . . . ; expressing, for every variable v and program statement s, the value taken by v when s is executed as a Boolean function V(s,v)(x, y, . . . ) of the bits of x, y, . . . ; and expressing, for every statement s, the condition under which the statement is executed as a Boolean function C(s)(x, y, . . . ) of the bits of the inputs x, y, . . . . Finally, a determination is made using a Boolean satisfiability oracle of whether, for the given variable v and program statement s, the following Boolean expression holds: C(s)(x,y, . . . )=&gt;V(s,v)(x,y . . . )=f(x,y, . . . ). In a second embodiment, the method includes expressing the value of f(x,y, . . . ) as a plurality of functions f j (x,y, . . . ) having the corresponding predicate P j (x,y, . . . ); expressing, for every variable v and program statement s, the value taken by v when s is executed as a plurality of functions V j (s,v)(x,y, . . . ), one for each predicate P j (x,y, . . . ); and expressing, for every statement s, the condition under which the statement is executed as a plurality of functions C j (s)(x,y, . . . ), one for each predicate P j (x,y, . . . ). Finally, a determination is of whether for the given variable v and program statement s, V j (s,v)(x,y, . . . )=f j (x,y, . . . ) whenever the predicate P j (x,y, . . . ) and the condition C j (s)(x,y, . . . ) are true.

CROSS REFERENCE

The present application claims the benefit of the U.S. provisional application Ser. No. 60/425,251 filed Nov. 12, 2002; is a continuation-in-part of U.S. patent application Ser. No. 10/382,578 filed Mar. 7, 2003; and claims the benefit of the U.S. provisional application Ser. No. 60/600,999 filed Aug. 12, 2004, all of which are incorporated herein by reference.

FIELD OF THE INVENTION

The present disclosure is directed to pattern recognition of instruction s in a program and more specifically to recognition of acyclic patterns by an optimizing compiler.

BACKGROUND AND SUMMARY OF THE INVENTION

Reference made herein and listed at the end are incorporated herein as necessary for understanding and implementing of the present disclosure.

Fixed-point saturating arithmetic is used widely in media and digital signal processing applications. Given a B-bit two's complement operation OP, its saturating counterpart detects overflow and sets the result to the limits of the representable number range:

$\begin{matrix} {{{OP}_{sat}\left( {x,y} \right)} = \begin{matrix} {{MaxB}\mspace{14mu} {if}\mspace{14mu} {{OP}\left( {x,y} \right)}\mspace{14mu} {overflows}} \\ {\left. {{MinB}\mspace{14mu} {if}\mspace{14mu} {OP}} \right\} \left( {x,y} \right)\mspace{14mu} {underflows}} \\ {{{OP}\left( {x,y} \right)}\mspace{14mu} {otherwise}} \end{matrix}} & (1) \end{matrix}$

For example, speech codecs in the GSM standard are expressed in terms of saturating arithmetic operations (Refs 1 and 2). A number of modern processor architectures, such as TI C6x, Intel IA-32 and ARM implement saturating arithmetic instructions.

Standard programming languages, such as ANSI C, do not provide operators for saturating arithmetic. Typically, application developers include libraries of functions or macros that implement these operations in terms of basic two's complement arithmetic. In order to obtain high performance on a given architecture, these libraries have to be coded either in assembly or using vendor-specific libraries or language extensions. This clearly complicates software development and maintenance. Various language extensions and standards have been proposed (Ref. 3). Their future is unclear, given the usual reluctance of established vendors to standardize. Thus, programmers are forced into the trade-off between portability and efficiency.

The way to resolve this trade-off is for a compiler to automatically recognize code fragments that implement saturating arithmetic and convert them into appropriate machine instructions. For this solution to be effective, the recognition algorithm must be robust and work across different coding styles. One can start with simple pattern matching on abstract syntax trees.

However, the limitations of such approach become evident as one consider various implementations of saturating addition. Examples are show in U.S. patent application Ser. No. 10/382,578. In order to prove, two saturated addition instructions are both equivalent to the definition in equation 1, one needs to prove equivalence between the results of the sequences of various bit manipulation and logical operations. Instead of an ad-hoc approach, a method that has a formal algebraic foundation is needed.

In the U.S. patent application Ser. No. 10/382,578, a method is described for recognition of saturated addition and subtraction specifically. The present disclosure recognizes that the principles in the previous application are applicable to any acyclic operation and may be expressed in generic terms.

The method of the present disclosure determines by an optimizing compiler whether any variable in the given program equals to the given acyclic mathematical function f(x,y, . . . ) applied to the given variables x, y, . . . in the program. In one embodiment, the method includes expressing the bits of the value of the function f(x,y, . . . ) as a Boolean function of the bits of the inputs x, y, . . . ; expressing, for every variable v and program statement s, the value taken by v when s is executed as a Boolean function V(s,v)(x, y, . . . ) of the bits of x, y, . . . ; and expressing, for every statement s, the condition under which the statement is executed as a Boolean function C(s)(x, y, . . . ) of the bits of the inputs x, y, . . . . Finally, a determination is made using a Boolean satisfiability oracle of whether, for the given variable v and program statement s, the following Boolean expression holds: C(s)(x,y, . . . )=>V(s,v)(x,y . . . ).

In a second embodiment, the method includes expressing the value of f(x,y, . . . ) as a plurality of functions f_(j)(x,y, . . . ) having the corresponding predicate P_(j)(x,y, . . . ); expressing, for every variable v and program statement s, the value taken by v when s is executed as a plurality of functions V_(j)(s,v)(x,y, . . . ), one for each predicate P_(j)(x,y, . . . ); and expressing, for every statement s, the condition under which the statement is executed as a plurality of functions C_(j)(s)(x,y, . . . ), one for each predicate P_(j)(x,y, . . . ). Finally, a determination is of whether for the given variable v and program statement s, V_(j)(s,v)(x,y, . . . )=f_(j)(x,y, . . . ) whenever the predicate P_(j)(x,y, . . . ) and the condition C_(j)(s)(x,y, . . . ) are true.

In either embodiment, the program may be transformed so that the value of each variable v that takes on the value of the function f(x,y, . . . ) is computed by the available function f(x,y, . . . ) by adding the invocation of the instruction t=f(x,y, . . . ); and replacing each reference to variable v at each statement s, such that the value of v at s equals f(x,y, . . . ), with the reference to t. The instructions that are unused after the program transformation are eliminated by dead-code elimination. Where the mathematical function f(x, y, . . . ) is computable in hardware, the program may be transformed so that the value of each variable v that takes on the value of the function f(x,y, . . . ) is computed by the available hardware instruction for f(x,y, . . . ).

These and other aspects of the present invention will become apparent from the following detailed description of the invention, when considered in conjunction with accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart of a first embodiment of a method incorporating the principles of the present disclosure.

FIG. 2 is a flow chart of a first embodiment of a method incorporating the principles of the present disclosure

FIG. 3 is a group of diagrams and data flow.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

A set of techniques will be described for automatic recognition of saturating arithmetic operations. In most cases the recognition problem is simply one of Boolean circuit equivalence. Given the expense of solving circuit equivalence, a set of practical approximations based on abstract interpretation is also described.

Experiments show that the present techniques, while reliably recognizing saturating arithmetic, have small compile-time overhead. The present method is not limited to saturating arithmetic, but is directly applicable to recognizing other idioms, such as add-with-carry and arithmetic shift.

Saturating operations can be defined by acyclic program fragments. Therefore, their results and intermediate values can be described bit-wise as Boolean expressions in terms of the inputs. This observation suggests the following solution:

Given an acyclic candidate code fragment, extract Boolean expressions that describe the values of its variables.

Using a Boolean expression equivalence oracle, test whether any of the expressions match the template.

To test the practicality of this method, a recognition algorithm for saturating addition employing a binary decision diagram package BuDDy (ref. 4) as the oracle was implemented. The algorithm picks a candidate two's complement addition operation z=x+y and attempts to discover which values in the program that depend on the operation carry its saturated value. For each edge in the control-flow graph of the program, the algorithm computes (whenever possible) the condition under which the edge is executed. The condition is expressed as a Boolean function over the bits of x, y. For each variable, the algorithm computes its value at every edge as a Boolean function over the bits of x, y.

An outline of the algorithm based on BDDs is:

ExactMatch (G: program CFG, OP: the operation to match) {repeat{

Pick candidate values x and y

if none found, exit

Let F be the Boolean expression for the bits of OP(x,y)

For every edge e and variable v:

-   -   Compute edge condition C(e)     -   Compute variable value V(e,v)

Identify program edges e and variables v such that:

∀x,y:C(e)(x,y)

(F(x,y)=V(e,v)(x,y))

-   -   (with Boolean expressions being tested via BDDs)

For every candidate addition operation OP(x, y), the algorithm forms the Boolean expression F(x,y) for the saturated value. It also expresses conditions C(e) under which edges e are executed and values of variables V(e,v) in terms of the inputs to the candidate operation. The algorithm then searches for program locations (edges) and variables that carry the saturated value, when they are executed. Computation of edge conditions and variable values can be phrased as a standard abstract interpretation problem (Refs. 5, 6 and 7). The domain of abstract interpretation in this case is the set of all Boolean functions over the candidate inputs. This method of embodiment one is shown in FIG. 1.

The time to detect saturating addition using this first embodiment is large—for instance, trying to prove an add was a saturating add required 10 milliseconds per attempt. Even after adding a heuristic that filtered out the majority of adds, using BDDs could add tens of seconds to the compile-time. An attempt was made to recognize saturating multiplication using this method, but ran out of memory while forming the BDD for the initial multiply. With improvements in hardware and/or Boolean expression equivalence or satisfiability oracle, the first embodiment can be used for saturating multiplication, for example.

Given the difficulty using the current hardware and Boolean solvers, a fast approximate algorithm to find saturating arithmetic operations was developed as a second embodiment. Approximations to the Boolean expressions that describe the values of program variables are constructed. By construction, the approximations are bounded in size and can be quickly tested for equivalence.

A given saturating operation (e.g. addition) can be implemented using a number of idioms (e.g. full-range, sub-range) and each idiom can be implemented in a number of ways. Each idiom can be described in terms of simple properties of statements and variables. This allows proving equivalence of program fragments without solving general Boolean satisfiability.

The problem of idiom recognition may be defined as follows:

-   -   given a statement that computes the two's complement operation:         z=xOPy     -   given the definition of the saturated value in terms of the         inputs x, y and the result z. The definition must be in the         predicated form:

${{OPsat}\left( {x,y,z} \right)} = \left\{ \begin{matrix} {{f_{1}\left( {x,y,z} \right)}\mspace{14mu}} & {{if}\mspace{14mu} {{Condition}_{1}\left( {x,y,z} \right)}} \\ {f_{2}\left( {x,y,z} \right)} & {{if}\mspace{14mu} {{Condition}_{2}\left( {x,y,z} \right)}} \\ \ldots & \; \end{matrix} \right.$

-   -   The goal is to identify values in the program that equal to         OP_(sat)(x,y,z). For example, full-range addition idiom can be         defined as:

${{addsat}\left( {x,y,z} \right)} = \left\{ {\begin{matrix} {MAX} & {{if}\mspace{14mu} {\left( {x \geq 0} \right)\bigwedge\left( {y \geq 0} \right)\bigwedge\left( {z < 0} \right)}} \\ {MIN} & {{if}\mspace{14mu} {\left( {x < 0} \right)\bigwedge\left( {y < 0} \right)\bigwedge\left( {z \geq 0} \right)}} \\ \ldots & \; \end{matrix}\begin{matrix} \mspace{14mu} \\ \mspace{14mu} \\ \; \end{matrix}} \right.$

In the L_add function from the GSM EFR codec shown below, the value of the L_var_outvariable at the return statement on line 14 matches this definition. If this can be proved automatically, then the return value can be computed using a saturating add instruction (if one is available on the target architecture):

 1: Word32 L_add(  2: Word32 L_var1, Word32 L_var2  3: ){  4: Word32 L_var_out;  5: L_var_out = L_var1 + L_var2;  6: if (((L_var1{circumflex over ( )}L_var2) & MIN_32)= =0)  7: {  8: if ((L_var_out{circumflex over ( )}L_var1) & MIN_32)  9: { 10: L_var_out = 11:  (L_var1 < 0) ? MIN_32 : MAX_32; 12: } 13: } 14: return NATIVE_ADDSAT(L_var1,L_var2); 15: }

Dead-code elimination would then produce the optimal code:

1: Word32 L_add( 2: Word32 L_var1, Word32 L_var2 3: ){ 4: return NATIVE_ADDSAT(L_var1, L_var2); 5: }

Initially, the input program has been lowered into three-address form represented as a Control-Flow Graph G=(N, E) of statements. The nodes include the unique start and end nodes, assignment statements of the form x:=OP (y, z), switches of the form IF(x≠0) and explicit control-flow merges.

If only scalar types available are 32-bit int and 16-bit short, the derivations can be easily generalized to operations of other bit-widths, such as 8-bit signed and unsigned pixel operations.

Given a program location and variables x and y, and the task is to identify variables v which on all executions of the program carry the value of the given function f(x,y). Formally, we are given:

-   -   A candidate program edge e₀ and candidate variables x, y.     -   A heuristic witness z. The witness helps simplify the         predicates. In the case of saturating operations, the witness is         usually the result of the non-saturating operation.     -   pattern function f(x,y,z) represented by a disjunction of         implications:

$\begin{matrix} {{f\left( {x,y,z} \right)} = \left\{ {\begin{matrix} {f_{1}\left( {x,y,z} \right)} & {{if}\mspace{14mu} {P_{1}\left( {x,y,z} \right)}} \\ {f_{2}\left( {x,y,z} \right)} & {{if}\mspace{14mu} {P_{2}\left( {x,y,z} \right)}} \\ \ldots & \; \\ {f_{n}\left( {x,y,z} \right)} & {{if}\mspace{14mu} {P_{n}\left( {x,y,z} \right)}} \end{matrix}\begin{matrix} \mspace{14mu} \\ {\mspace{11mu} \mspace{11mu}} \end{matrix}} \right.} & (2) \end{matrix}$

The goal is to identify such edges e and variables v, that the value of v when e is executed is f(x₀,y₀,z₀) where x₀, y₀ and z₀ are the values of the candidate variables at the most recent execution of the candidate edge e₀.

In this embodiment for a given candidate edge and candidate variables, perform abstract interpretation once for each condition in equation 2. During each successive instance of abstract interpretation, compute approximations to the variable values V(e,a) and edge conditions or assertions A(e) under the assumption that the predicate P_(j) holds true.

An edge assertion A(e) is the condition that holds true if the program edge is executed. The edge assertions A(e) is computed as functions of the candidates x, y and the witness z. Variable value V(e,v) represents knowledge about the value of v when the edge e is executed. V(e,v) is also a function of x, y and z.

The program is modified to save at the candidate edge e₀ the values of x, y and z into the temporaries x₀, y₀, z₀. Then at every edge e the assertion assert (A(e)(x₀,y₀,z₀)) and, for every variable v, the assertion assert(v==V(e,v)(x₀,y₀,z₀)) will be satisfied.

Given the above definitions, the variable α carries the value of the pattern function f(x,y,z), if the following holds:

∀j:∀x,y,z:(A(e)(x,y,z)

P _(j)(x,y,z))

(V(e,a)(x,y,z)=f _(j)(x,y,z))  (3)

In other words, for each condition P_(j) in equation 2 is tested either (a) the edge e is not executed, or (b) the edge is executed and the variable a equals to the appropriate case f_(j). The second embodiment is illustrated in FIG. 2.

To approximate the assertions and the values, two lattices are used: one to approximate conditions or predicates and one to approximate values. Φ denotes the lattice of predicates and Λ denotes the lattice of values. The lattices must satisfy the following requirements:

-   -   Both Φ and A must include the top and bottom elements. Both must         define the meet operator and the comparison operator. The         comparison operator is necessary for a well-formed lattice. The         meet operator is used at program merge points.     -   Φ must define the conjunction, disjunction and negation. Φ must         contain both the true value and false value. Φ must be sound in         that the usual algebraic rules for the conjunction, disjunction,         truth and falsehood must hold.     -   For every arithmetic operation OP in the intermediate program         language, the lattice Λ must define an interpretation OP: Λ×Λ→Λ.         (we assume that all operations are binary.)

Two functions that link the values to the respective predicates:

-   -   The MayNonZero: Λ→Φ function tests whether a value can possibly         be non-zero.     -   The MayZero: Λ→Φ function tests whether a value possibly be         zero.

The MayNonZero and MayZero functions do not have to produce contradictory results. Sometimes it might be necessary to approximate a value as possibly being zero and possibly being non-zero at the same time.

Given these definitions, abstract interpretation of a program in terms of the edge assertions A(e) and variable values V(e,a) relative to the candidate edge e₀, the candidate variables x, y and the witness z is defined. The equations are shown in FIG. 3. This is a fairly standard set of equations. The equations for merge statements allow for Wegman-Zadeck style one-side conditional propagation (Refs 8 and 9).

The overall algorithm is summarized as follows:

ApproxMatch ( G : program CFG, OP : the operation to match ) \{ repeat \{ Heuristically pick candidate values x, y and the witness z if none found, exit For each implication Pj(x,y,z)

fj (x,y,z) (in the pattern function) { Assume Pj is true Compute edge assertions A(e) and variable values V(e,v) using abstract interpretation Identify edges e and variables v that satisfy: ∀ x, y, z : (A(e)(x,y,z)

P_(j)(x,y,z))

(V(e,a)(x,y,z) = f_(j)(x,y,z)) \}\}\}

Examples of how the present methods applies to multiplication and full-range and sub-range addition follow. It is straight-forward to extend our techniques to recognition of saturating subtraction. Further examples to be illustrated includes the recognition of add with carry ane arithmetic shift. The method is also applicable to operations such as clipping.

A. Full-Range Saturating Addition

To match full-range saturating addition, start with a non-saturating addition operation z=x+y. The input variables x and y are the candidate variables. The output variable z is the witness. The pattern function is:

$\begin{matrix} {{f\left( {x,y,z} \right)} = \left\{ {\begin{matrix} {{MAX}\; 32} & {{{if}\mspace{14mu} x} + {y\mspace{14mu} {overflows}}} \\ {{MIN}\; 32} & {{{if}\mspace{14mu} x} + {y\mspace{14mu} {underflows}}} \\ z & {otherwise} \end{matrix}\begin{matrix} \mspace{14mu} \\ \; \end{matrix}} \right.} & (4) \end{matrix}$

The overflow and underflow predicates can be defined through sign tests:

$\begin{matrix} {{f\left( {x,y,z} \right)} = \left\{ \begin{matrix} {{MAX}\; 32} & {{if}\mspace{14mu} {\left( {x \geq 0} \right)\bigwedge\left( {y \geq 0} \right)\bigwedge\left( {z < 0} \right)}} \\ {{MIN}\; 32} & {{if}\mspace{20mu} {\left( {x < 0} \right)\bigwedge\left( {y < 0} \right)\bigwedge\left( {z \geq 0} \right)}} \\ z & {otherwise} \end{matrix} \right.} & (5) \end{matrix}$

The comparison operators in this definition use 32-bit two's complement arithmetic.

Various implementations of full-range saturating addition either branch on predicates in terms of the sign bits of x, y and z, and/or perform bit manipulation of the sign bits. Therefore, edge assertions are represented as predicates of the sign bits (and more). Additionally, keep track of whether the section [p,p+n] of bits of any variable v is equal to the section [q,q+n] of bits of x, y or z.

The lattice Φ is the set of all predicates in terms of the six input variables:

$\begin{matrix} {{\sigma_{x} \equiv {\left( {x < 0} \right)\mspace{14mu} {the}\mspace{14mu} {sign}\mspace{14mu} {bit}\mspace{14mu} {of}\mspace{14mu} x}}\text{}{\sigma_{y} \equiv {\left( {y < 0} \right)\mspace{14mu} {the}\mspace{14mu} {sign}\mspace{14mu} {bit}\mspace{14mu} {of}\mspace{14mu} y}}{\sigma_{z} \equiv {\left( {z < 0} \right)\mspace{14mu} {the}\mspace{14mu} {sign}\mspace{14mu} {bit}\mspace{14mu} {of}\mspace{14mu} z}}\text{}{\zeta_{x} \equiv \left( {x = 0} \right)}{\zeta_{y} \equiv \left( {y = 0} \right)}{\zeta_{z} \equiv \left( {z = 0} \right)}} & (6) \end{matrix}$

The value lattice Λ=Φ³²×Γ is built as product of two lattices:

-   -   Φ³² is the lattice of all vectors of length 32 of predicates in         Φ. Φ³² is used to approximate the bits of variables in terms of         the sign bits of x, y and z.     -   Γ={T x_(p,q,n), y_(p,q,n)}, z_(p,q,n), ⊥} is used to approximate         variables as sub-ranges of bits of x, y and z. The approximation         of a variable v as x_(p,q,n) means that the bits v[p,p:n] of v         are same as the bits x[q,q+n] of x.

Each element in Λ is a pair

ν,ρ

where

ν,ρ

where ν ε Φ³² and ρ ε Γ. ν is the vector component and ρ is the bit range component.

The interpretations OP of arithmetic operators in terms of the vector components are rather straight-forward, namely for any arithmetic operation, a predicate can be derived for each bit of the result. Heuristically, do not interpret multiplication, division and remainder operations, other than the ones that can be converted to a shift or bit-wise. The results of these operations are (conservatively) set to vectors of ⊥.

The interpretations in terms of bit range components are rather simple, too. Constant shifts, and bit-wise operations with a constant produce outputs that are not ⊥. All other operations are not interpreted.

As an example of interpretation, consider the equality comparison: c:=(a=0). If the bit range component of α is of the form ν_(0, 0, 31) for ν ε x, y, z, then the vector component of c must be set to:

0 . . . 0 ζν:

.

The MayNonZero and MayZero operators test whether representation of the input value can be non-zero or zero, respectively. MayNonZero forms the disjunction of all the bits in the Φ³² component. If the disjunction is not trivially false, then MayNonZero returns 1. Similarly, if the disjunction is not trivially true, then MayZero returns 1.

The conditions P_(j) for the pattern function are trivially encoded:

$\begin{matrix} {{{{Overflow}\left( {x,y,z} \right)} \equiv {{\sigma_{x}\bigwedge{{\sigma_{y}\bigwedge{\sigma_{z}}}}}}}{{{Underflow}\left( {x,y,z} \right)} \equiv {\sigma_{x}\bigwedge\sigma_{y}\bigwedge\sigma_{z}}}{{{NoOverflow}\left( {x,y,z} \right)} \equiv {{{Overflow}\bigwedge{{Underflow}}}}}} & (7) \end{matrix}$

The initial values for the candidate edge e₀ are:

V(e ₀ ,x)=

σ_(x) ⊥ . . . ⊥

x _(0,0,31)

V(e ₀ ,y)=

σ_(x) ⊥ . . . ⊥

y _(0,0,31)

V(e ₀ ,z)=

σ_(x) ⊥ . . . ⊥

z _(0,0,31)

  (8)

To illustrate how abstract interpretation works in this instance, consider the following example:

 1: int addsat_32bit (int x, int y) {  2: int z = x + y;  3: int t1 = ( ~(x {circumflex over ( )} y) & (x {circumflex over ( )} z) );  4: int t2 = MIN32 & t1;  5: if ( t2 ) {  6: int t3 = (x >> 31) & 1;  7: z = MAX32 + t3;  8: }  9: return z; 10: } Only the most-significant bit of t1 (line 3) can be represented:

V1=

(σ_(x)⊕σ_(y)))

(σ_(x)⊕σ_(z))), ⊥, . . . , ⊥

, ⊥

Conjunction with MIN32 (i.e. 0x80000000) isolates the most significant bit of t1. Thus the representation of the value of t2 on line 4 is:

V2=

(σ_(x)⊕σ_(y)))

(σ_(x)⊕σ_(z))), 0, . . . , 0

, ⊥

The condition under which statements on lines 6 and 7 are executed is given by the disjunction of the bits of t2, since the implicit test is t2≠0:

MayNonZero(V ₂)=(

(σ_(x)⊕σ_(y)))

(σ_(x)⊕σ_(z)))

Therefore, the assertion for the lines 6 and 7 is:

$\begin{matrix} {A_{1} = {P\bigwedge{{MayNonZero}\left( V_{2} \right)}}} \\ \left. {P\bigwedge\left( {\left( {\sigma_{x} \oplus \sigma_{y}} \right)} \right)\bigwedge\left( {\sigma_{x} \oplus \sigma_{z}} \right)} \right) \end{matrix}$

where P is the current assumption (overflow, underflow, neither) as given by equation 7. It is easy to see that A₁=0 if we assume “neither” and it is non-zero otherwise. Thus, under the “neither” assumption only the assignment to z at the candidate edge reaches the return statement (line 9).

The computed approximation for t3 on line 6 is:

V₃=

0,0 . . . 0,σ_(x)

, x_(0,31,0)

I.e. t3 carries the sign (31st) bit of x. σ_(x)=0 under the assumption of overflow. Therefore, z is assigned the constant MAX32. When underflow is assumed, σ_(x)=1 and z is assigned MAX32+1==MIN32.

All of the above proves that the return statement is reached by the value MAX32 when overflow occurs, MIN32 when underflow occurs and the original value of z, otherwise. This, in turn, proves that the return statement is reached by the saturated value.

B. Sub-Range Addition

The pattern function for sub-range addition is:

$\begin{matrix} {{f\left( {x,y,z} \right)} = \left\{ \begin{matrix} {{{MAX}\; 16\mspace{14mu} {if}\mspace{14mu} z} > {{MAX}\; 16}} \\ {{{MIN}\; 16\mspace{14mu} {if}\mspace{14mu} z} < {{MIN}\; 16}} \\ {z\mspace{14mu} {otherwise}} \end{matrix} \right.} & (9) \end{matrix}$

The comparisons in this definition use 32-bit two's complement arithmetic.

Sub-range addition is simpler. The candidate addition z=x+y where x and y are declared as 16-bit “shorts”. Overflow, underflow and no-overflow conditions are represented in terms of range tests on z:

Overflow(x, y, z) ≡ (z∈]Max 16, +∞[) Underflow(x, y, z) ≡ (z∈] − ∞, Min 16[NoOverflow(x, y, z) ≡ (z ∈ [Min 16, Max 16]

Therefore, range propagation is performed for the overflow and underflow assumptions and propagate the property “this variable equals to z” for the no-overflow assumption.

C. Multiplication

Consider an implementation of fixed-point saturating multiplication (16-bit into 32-bit):

 1: #define MAX32 (0x80000000)  2: int mult_sat (short x, short y) {  3: int z = x * y;  4: if (z == 0x40000000) {  5: z = MAX32;  6: }  7: else {  8: z = 2*z;  9: } 10: return z; 11: }

In fact, a more general idiom is that of “multiplication shifted by k”:

 1: #define MAX32 (0x80000000)  3: #define k (.....)  2: int mult_sat_k (short x, short y) {  3: int z = x * y;  4: if (z == 0x40000000) {  5:  z = MAX32;  6: } 10: return (z >> (k−1)); 11 } This computes the product of x and y represented as a fixed-point number in 32-k bits. Typically k=16.

In order to determine what values are assigned when overflow occurs, simply perform constant propagation with the additional mappings defined at the

V(e ₀ ,x)=0x8000

V(e ₀ ,y)=0x8000

V(e ₀ ,z)=0x40000000

In order to determine which values are assigned when no overflow occurs, keep track of which constant (if any) a variable equals to, which constant it is known not to be equal to and whether a variable contains a range of bits from x, y and z. The value lattice in this case is the product:

Λ=C×C×Γ

where the lattice Γ has the same meaning as in example A. It is used to keep track of bit ranges. C is the lattice commonly used in constant propagation. The C lattice contains the same elements as C, but it has the inverted meaning: a variable a is assigned const ε C if it is known never to be equal to the constant.

Interpretation for the arithmetic operations is straight-forward. Special care must be taken to define rules for interpretation of comparisons. The equality cmpeq test is interpreted via the following rules:

${{cmpeq}\left( {{\langle{{c_{1,}\_},\_}\rangle},{\langle{{c_{2,}\_},\_}\rangle}} \right)} = \left\{ {{\begin{matrix} {{{\langle{1,{\bot{,\bot}}}\rangle}\mspace{14mu} {if}\mspace{14mu} c_{1}} = c_{2}} \\ {{{\langle{0,{\bot{,\bot}}}\rangle}{\mspace{11mu} \;}{if}\mspace{14mu} c_{1}} \neq c_{2}} \end{matrix}{{cmpeq}\left( {{\langle{{c_{,}\_},\_,}\rangle},{\langle{\_,c,\_}\rangle}} \right)}} = {\langle{0,{\bot{,\bot}}}\rangle}} \right.$

Similar rules are defined for other comparisons. A special rule for multiplication is:

mul(

0x8000,_,

, 0x8000,_

) =

_, 0x8000,_

)=

_(—),0x40000000,_

I.e. if we multiply 0x8000 with something that is not 0x8000, we can never get 0x40000000.

Special care must also be taken not to create contradictory representations such as

0,0,_

: never create a tuple with both the element of C and that of C being set to a constant. If the latter is set to a constant, set the former to ⊥.

In order to simplify analysis, divide the definition of shifted saturating multiplication into three cases, two of which handle the non-overflow condition, giving the pattern function:

${f\left( {x,y,z} \right)} = \left\{ \begin{matrix} {z\left( {k - 1} \right)} & {{{if}\mspace{14mu} x} \neq {0 \times 8000}} \\ {z\left( {k - 1} \right)} & {{{if}\mspace{14mu} y} \neq {0 \times 8000}} \\ {{{Max}\; 32}k} & {{{if}\mspace{14mu} x} = {y = {0 \times 8000}}} \end{matrix} \right.$

In the first case, abstract interpretation if performed with the initial value of x at the candidate edge being

⊥,0x8000, x_(0,0,15)

. In the second case, similar lattice value is given to y. In both cases, the witness z is assigned

⊥,0x40000000, Z_(0,0,31)

. In the third, overflow, case start with the assignment:

V(e ₀ ,x)=

0x8000,⊥,x _(0,0,15)

V(e ₀ ,y)=

0x8000,⊥,x _(0,0,15)

V(e ₀ ,z)=

0x40000000,⊥,z _(0,0,31)

D. Add With Carry

Given an addition operation z=x+y and a carry-in variable c_(in), identify variables that hold the carry-out bit. Identifying the result of the addition is quite simple: use global value numbering to identify variables that hold z+c_(in). An important assumption is that from prior program analysis (such as range propagation), that c_(in) ε {0,1}. An example is:

 1: int addc(int x, int y,  2: int c_in, int *p_c_out){  3: int z = x + y;  4: int c_internal;  5: int c_out;  6: assert (c == 0 ∥ c == 1);  7: if ((x < 0) && (y < 0)) {  8: c_internal = 1;  9: } 10: else if (MIN32 & ((x {circumflex over ( )} y) & ~z)) { 11: c_internal = 1; 12: } 13: else { 14: c_internal = 0; 15: } 16: if (z == 0xFFFFFFFF) { c_out = c_in; } 17: else { c_out = c_internal; } 18: *p_c_out = c_out; 19: return (z + c_in); 20:}

To define the pattern function for the carry-out bit, first define the “carry internal” bit, as:

Carrylnternal(x,y,z)=≡(σ_(x)

σ_(y))

((σ_(x)⊕σ_(y))

σ_(z)

This is the carry-out bit in the addition $z=x+y$, without the presence extra carry-in bit. In the presence of the carry-in bit, the carry-out function is defined as:

${{Carry}\left( {x,y,z,c_{i\; n}} \right)} = \left\{ \begin{matrix} {{c_{i\; n}\mspace{14mu} {if}\mspace{14mu} z} = {0 \times {FFFFFFFF}}} \\ {{CarryInternal}\left( {x,y,z} \right)} \\ {otherwise} \end{matrix} \right.$

This is the pattern function. Re-use some of the process of the previous sections in order to detect carry-out computations. Abstract interpretation is performed under the assumptions:

z = 0 × FFFFFFFF z ≠ 0 × FFFFFFFF

The “equals-to-constant” and “not-equals-to-constant” information needs to be propagated. We utilize the C and C lattices that were introduced for detection of saturating multiplication. Detect computations of CarryInternal, which is a Boolean predicate in the sign bits σ_(x),σ_(y),σ_(z) and the carry-in bit c_(in). For this we utilize the Φ³². Overall, the value lattice is

=C×C×Φ³².

Observe that the condition z=0xFFFFFFFF also implies that x and y must have different signs. So the abstract interpretation for the first case, is performed under the assumption:

P ₁≡(σ_(x)⊕σ_(y))

σ_(z)

At the candidate edge, the variable z is assigned the lattice value:

V(e ₀ ,z)=

0xFFFFFFFF,⊥,

1,1, . . . , 1

To implement the z≠0xFFFFFFFF assumption, assign z the initial lattice value:

V(e ₀ ,z)=

⊥0xFFFFFFFF,

σ_(z),⊥, . . . , ⊥

E. Arithmetic Shift Computation.

ANSI C standard does not define which bits the right shift operator “x>>y” brings in from the left when the first operand is of a signed type. The question is whether it brings in the sign bit of x or zero. In the first case, it would be called “arithmetic shift”, in the second—“logical shift”.

Here are a couple of portable implementations of arithmetic shift:

int arith_shift_1 (int x, int y) { if (y < 0) { return x << (−y); } else if (y >= 31) { return (x < 0) ? −1 : 0; } else { int z = (x >> y); int t = (1 << (31 − y)); if (z & t) { z = z | (0xFFFFFFFF << (32 − y)); } return z; } } int arith_shift_2 (int x, int y) { if (y < 0) { return x << (−y); } else if (y >= 31) { return (x < 0) ? −1 : 0; } else { if (x < 0) { return ~(~x >> y); } else { return x >> y; } } } Observe that (a) variables get shifted by the amount that is linear in y, (b) most of the values are divided into two bit ranges [0:p(y)] and [p(y)+1,31] where p(y) is a linear function of y. The values of the two bit ranges are either constant, or a sub-range of x, possibly inverted.

Divide the definition of arithmetic shift into 5 cases:

${{Shra}\left( {x,y} \right)} = \begin{matrix} \left( {x\left( {- y} \right)} \right) & {{{if}\mspace{14mu} y} < 0} \\ {- 1} & {{{if}\mspace{14mu} y} \geq {31\bigwedge x} < 0} \\ 0 & {{{if}\mspace{14mu} y} \geq {31\bigwedge x} \neq {0 \smallsetminus}} \\ \left( {xy} \right) & {{{if}\mspace{14mu} 0} \leq y \leq {30\mspace{14mu} {and}\mspace{14mu} x} \geq 0} \\ {\left( {{xy}} \right)} & {{{if}\mspace{14mu} 0} \leq y \leq {30\mspace{14mu} {and}\mspace{14mu} x} < 0} \end{matrix}$

(The sign-extension behavior of the “native” right shift operator in the above formula is irrelevant.)

In order to determine which statements are executed for each of the five cases above, propagate ranges and the “equals-to-constant” and “not-equals-to-constant” properties. In order to determine which variables equal to x<<(−y), −1, 0, x>>y or

(

x>>y), perform global value numbering. Observe that one is just reusing the value lattices that were either introduced earlier or are expected to be implemented in a mature compiler.

The present methods allow the compiler to substitute a hardware implementation, where available for the software implementation of the operator when identity has been recognized. The program may be transformed so that the value of each variable v that takes on the value of the function f(x,y, . . . ) is computed by the available function f(x,y, . . . ) by adding the invocation of the instruction t=f(x,y, . . . ); and replacing each reference to variable v at each statement s, such that the value of v at s equals f(x,y, . . . ), with the reference to t. The instructions that are unused after the program transformation are eliminated by dead-code elimination. Where the mathematical function f(x, y, . . . ) is computable in hardware, the program may be transformed so that the value of each variable v that takes on the value of the function f(x,y, . . . ) is computed by the available hardware instruction for f(x,y, . . . ).

The above techniques for automatic recognition of saturating arithmetic operations have fairly low compile-time overhead. Further, these techniques are fairly general in that they recognize underlying saturating arithmetic operations even when expressed using considerably different idioms. The generality of this approach arises from the fact that instead of doing a syntactic pattern match, the underlying semantic concept is exploited in a systematic fashion. Further, abstract interpretation allows application of this reasoning in systematically, as opposed to ad-hoc fashion. Once the appropriate semantic elements have been determined and the lattices constructed, the interpretation for each operation can be systematically determined.

Although the present invention has been described and illustrated in detail, it is to be clearly understood that this is done by way of illustration and example only and is not to be taken by way of limitation. The scope of the present invention are to be limited only by the terms of the appended claims.

REFERENCES

-   [1] E. T. S. Institute, Digital cellular telecommunications system,     ANSI-C code for the GSM Enhanced Full Rate (EFR) speech codec. (GSM     46.053), ftp://ftp.3gpp.org/specs/archive. -   [2] ———, Digital cellular telecommunications system, ANSI-C code for     the GSM Adaptive Multiple Rate (AMR) speech codec. (GSM 26.073),     ftp://ftp.3gpp.org/specs/archive. -   [3] DSP-C Proposed Standard, www.dsp-c.org. -   [4] J. Lind-Nielsen, BuDDy—A Binary Decision Diagram Package,     http://www.itu.dk/research/buddy -   [5] P. Cousot and R. Cousot, “Abstract interpretation: a unified     lattice model for static analysis of programs by construction or     approximation of fixpoints,” in POPL 1979. -   [6]———, “Systematic design of program transformation frameworks by     abstract interpretation,” in Proceedings of the 29th annual ACM     SIGACT-SIGPLAN symposium on Principles of programming languages. ACM     Press, 2002, pp. 178-190. -   [7]———, “Automatic synthesis of optimal invariant assertions:     Mathematical foundations,” in Proceedings of the 1977 symposium on     Artificial intelligence and programming languages, 1977, pp. 1-12. -   [8] M. N. Wegman and F. K. Zadeck, “Constant propagation with     conditional branches,” TOPLAS, vol. 13, no. 2, pp. 181-210, 1991.

[9] K. Pingali, M. Beck, R. Johnson, M. Moudgill, and P. Stodghill, “Dependence flow graphs: an algebraic approach to program dependencies,” in Proceedings of the 18th ACM SIGPLAN-SIGACT symposium on Principles of programming languages. ACM Press, 1991, pp. 67-78. 

1. A method of determining by an optimizing compiler whether any variable in the given program equals to the given mathematical function f(x,y, . . . ) applied to the given variables x, y, . . . in the program, the method comprising: expressing the bits of the value of the function f(x,y, . . . ) as a Boolean function of the bits of the inputs x, y, . . . ; expressing, for every variable v and program statement s, the value taken by v when s is executed as a Boolean function V(s,v)(x, y, . . . ) of the bits of x, y, . . . ; expressing, for every statement s, the condition under which the statement is executed as a Boolean function C(s)(x, y, . . . ) of the bits of the inputs x, y, . . . ; and determining using a Boolean satisfiability oracle whether, for the given variable v and program statement s, the following Boolean expression holds: C(s)(x,y, . . . )=>V(s,v)(x,y . . . )=f(x,y, . . . ).
 2. The method according to claim 1, wherein the mathematical function f(x, y, . . . ) is computable in hardware
 3. The method according to claim 2, wherein the program is transformed so that the value of each variable v that takes on the value of the function f(x,y, . . . ) is computed by the available hardware instruction for f(x,y, . . . ):
 4. The method according to claim 3, including: adding the invocation of the hardware instruction t=f(x,y, . . . ); and replacing each reference to variable v at each statement s, such that the value of v at s equals f(x,y, . . . ), with the reference to t.
 5. The method according to claim 4, wherein the software instructions that are unused after the program transformation are eliminated by dead-code elimination.
 6. The method according to claim 1, wherein the program is transformed so that the value of each variable v that takes on the value of the function f(x,y, . . . ) is computed by the available function f(x,y, . . . ) by adding the invocation of the instruction t=f(x,y, . . . ); and replacing each reference to variable v at each statement s, such that the value of v at s equals f(x,y, . . . ), with the reference to t.
 7. The method according to claim 6, wherein the instructions that are unused after the program transformation are eliminated by dead-code elimination.
 8. A method of determining by an optimizing compiler whether any variable in the given program equals to the given mathematical function f(x,y, . . . ) applied to the given variables x, y, . . . in the program, the method comprising: expressing the value of f(x,y, . . . ) as a plurality of functions f_(j)(x,y, . . . ) having the corresponding predicate P_(j)(x,y, . . . ); expressing, for every variable v and program statement s, the value taken by v when s is executed as a plurality of functions V_(j)(s,v)(x,y, . . . ), one for each predicate P_(j)(x,y, . . . ); expressing, for every statement s, the condition under which the statement is executed as a plurality of functions C_(j)(s)(x,y, . . . ), one for each predicate P_(j)(x,y, . . . ); and determining whether for the given variable v and program statement s, V_(j)(s,v)(x,y, . . . )=f_(j)(x,y, . . . ) whenever the predicate P_(j)(x,y, . . . ) and the condition C_(j)(s)(x,y, . . . ) are true.
 9. The method according to claim 8, wherein the mathematical function f(x, y, . . . ) is computable in hardware
 10. The method according to claim 9, wherein the program is transformed so that the value of each variable v that takes on the value of the function f(x,y, . . . ) is computed by the available hardware instruction for f(x,y, . . . ).
 11. The method according to claim 10, including: adding the invocation of the hardware instruction t=f(x,y, . . . ); and replacing each reference to variable v at each statement s, such that the value of v at s equals f(x,y, . . . ), with the reference to t.
 12. The method according to claim 10, wherein the software instructions that are unused after the program transformation are eliminated by dead-code elimination
 13. The method according to claim 1, wherein the program is transformed so that the value of each variable v that takes on the value of the function f(x,y, . . . ) is computed by the available function f(x,y, . . . ) by adding the invocation of the instruction t=f(x,y, . . . ); and replacing each reference to variable v at each statement s, such that the value of v at s equals f(x,y, . . . ), with the reference to t.
 14. The method according to claim 13, wherein the instructions that are unused after the program transformation are eliminated by dead-code elimination.
 15. The method according to claim 8, wherein depending on the predicate P_(j)(x,y, . . . ) or the function f_(j)(x,y, . . . ), the compiler uses an approximation of the values of the program to determine whether a condition C_(j)(s)(x,y, . . . ) holds and whether V_(j)(s,v)(x,y, . . . )=f_(j)(x,y, . . . ) whenever P_(j)(x,y, . . . ) holds.
 16. The method according to claim 15, wherein the approximation is a combination of one or more of constant propagation, global value numbering, range propagation applied either to whole variables or to bit-ranges of variables. 